## anonymous 4 years ago calculate the integral x/[(x-a)(x-b)] when a=b

1. TuringTest

With a=b we can write$\int\frac x{(x-a)^2}dx$now a little u substitution$u=x-a\to x=u+a\to du=dx$$\int\frac{u+a}{u^2}du=\int u^{-1}+au^{-2}du$Got it from here?

2. anonymous

but shouldnt we split it into partial fractions first??

3. TuringTest

I don't see a need to...

4. TuringTest

but sure, that should give the same result seems like an unnecessary pain though

5. anonymous

but i think that is what they want here and that is where i am lost

6. TuringTest

$\int\frac x{(x-a)^2}=\int\frac A{x-a}+\frac B{(x-a)^2}dx$$A(x-a)+B=x\to Ax+B-Aa=x$so we get$A=1$$B-a=0\to B=a$so our integral is$\int\frac1{x-a}+\frac a{(x-a)^2}dx$looks to me like we're gonna get the same answer both ways, which is a good thing :D

7. anonymous

thank you